Though he is not expected to become Taoiseach until June 13, sources said Varadkar intends to engage in “talks in the near future about the transition process”.

Supporters of Varadkar warned: “Leo is facing some of the most serious issues in the history of the State — Brexit, the North, a Garda force that is out of control. He will need to be running before he hits the ground.”

They added: “There will be no honeymoon, not even a bank holiday weekend.”

But Varadkar will have to wait until the end of the week for any meeting with the long-serving outgoing Taoiseach as Enda is flying out today for a two-day trade and investment mission to Chicago.

He is then travelling on from America to Belgium where he will attend Messines Ridge commemorations on Wednesday.

Varadkar, who is already under pressure from internal and external allies and opponents, is engaged in a round-robin series of meetings and preparatory work to ensure he is elected as Taoiseach on June 13.

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The youthful Fine Gael head honcho is facing into tough negotiations with Fianna Fail and the Independent Alliance when it comes to the continuation of the government.

Speaking to the Irish Sun yesterday, Fianna Fail leader Micheal Martin repeated his initial congratulations to “Leo and his family’’ and said it was his intention to “continue to work the confidence and supply agreement’’.

However, Martin warned: ‘‘Fine Gael’s internal obsession has meant this Government has not been productive, we are deeply concerned about the inertia of the last year.”

For his party to retain confidence in the Coalition, he said: ‘‘The next 12 months must be about substance, there simply has to be delivery.”

In particular Martin added: “The new Taoiseach must convince us he has a handle on Brexit, on Justice and on Health.

“We have been promised things like the return of the Treatment Fund and nothing has happened. To end the scenario where more children are seeking help from the Capuchins than in 2013.”

Another senior Fianna Fail figure also warned: “There is unease in FF that FG are doing the same thing to us as they did to Labour, they make all the mistakes and even though we are doing right for the country we shoulder the blame.”

However, at his first press conference, Varadkar sent a strong signal that the Tanaiste’s post would not be on the table with the claim that Frances Fitzgerald was doing “a very good job” in the brief.

Sources told the Irish Sun on Sunday that the secret summit with Coveney was “a priority, we need to bring the party together again”.

Leo will also be starting the delicate political task of reuniting the party.

The victory of Coveney amongst the grassroots members has put Varadkar under pressure to ensure there is no Cabinet clear-out.

It is believed Leo will conduct a very small re-shuffle. The instinctively cautious Varadkar has remained tight-lipped about his plans with sources noting “even ‘Murph’ [Eoghan Murphy} doesn’t know if he is getting the leg up”.

Varadkar supporters, however, warned: “Leo is going to be completely on the back-foot if there is not an Albert style clear-out.”